Electronic Journal of Differential Equations, Vol. 2009(2009), No. 29, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OSCILLATION CRITERIA FOR FIRST-ORDER SYSTEMS OF
LINEAR DIFFERENCE EQUATIONS
ARUN KUMAR TRIPATHY
Abstract. In this article, we obtain conditions for the oscillation of vector
solutions to the first-order systems of linear difference equations
x(n + 1) = a(n)x + b(n)y
y(n + 1) = c(n)x + d(n)y
and
x(n + 1) = a(n)x + b(n)y + f1 (n)
y(n + 1) = c(n)x + d(n)y + f2 (n)
where a(n), b(n), c(n), d(n) and fi (n), i = 1, 2 are real valued functions defined
for n ≥ 0.
1. Introduction
Consider the system of k-equations of the form
X(n + 1) = AX(n),
(1.1)
where A = (aij )k×k is a constant matrix. The characteristic equation of (1.1) is
given by
det(λI − A) = 0;
that is,
λk + (−1)k b1 λk−1 + · · · + (−1)k bk = 0,
(1.2)
where bk = det A. If k is odd, then from the theory of algebraic equations (see e.g.
[2]), it follows that (1.2) admits at least one real root λ1 such that the sign of λ1
is opposite to that of the last term, namely (−1)k bk . Hence we have the following
result.
Theorem 1.1. Let k be odd. If det A < 0, then (1.1) admits at least one oscillatory
solution; if det A > 0, then (1.1) admits at least one nonoscillatory solution.
Proof. When det A < 0, we find a real root λ1 of (1.2) such that λ1 < 0 and
X(n) = (λ1 )n C, where C = (C1 C2 . . . Ck )T is a column vector of constants. Thus
X(n) is oscillatory. Similarly for det A > 0.
2000 Mathematics Subject Classification. 39A10, 39A12.
Key words and phrases. Oscillation; linear system; difference equation.
c
2009
Texas State University - San Marcos.
Submitted November 29, 2008. Published February 9, 2009.
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A. K. TRIPATHY
EJDE-2009/29
Remark. If det A = 0, then (1.1) admits a nonoscillatory solution. Indeed, det A =
0, implies that λ = 0 is a solution of (1.2) and hence X(n) = C is a solution of
(1.1), where C is a non-zero constant vector. We note that AC = 0 always admits
a nontrivial solution.
Theorem 1.2. Let k be even. If det A < 0, then (1.1) admits an oscillatory
solution and a nonoscillatory solution.
The proof is simple and can be obtained from the following Theorem in [2].
Theorem 1.3.
(I) Every equation of an even degree, whose constant term is
negative has at least two real roots one positive and the other negative.
(II) If the equation contains only even powers of x and the coefficients are all
of the same sign, then the equation has no real root; that is, all roots are
complex.
Remarks. If the last term of an even degree equation is positive, no definite
conclusion can be drawn regarding the roots of the equation. If det A > 0, then
no definite conclusion can be drawn regarding the oscillation of solutions of (1.1)
when k is even.
Theorem 1.4. Let k be even and A be such that b1 = b3 = · · · = bk−1 = 0,
b2 > 0, b4 > 0 . . . bk > 0. Then every component of the vector solution of (1.1) is
oscillatory.
The proof of the above theorem follows from the above Theorem 1.3(II).
The literature on study of system of difference equations does not conisder the
case when k is even. Therefore the present work is devoted to study the system of
equations
x(n + 1) = a(n)x + b(n)y
(1.3)
y(n + 1) = c(n)x + d(n)y
and the corresponding nonhomogeneous system of equations
x(n + 1) = a(n)x + b(n)y + f1 (n)
(1.4)
y(n + 1) = c(n)x + d(n)y + f2 (n),
where a(n), b(n), c(n), d(n), f1 (n), f2 (n) are real-valued functions defined for n ≥
n0 ≥ 0. One may think of systems (1.3) and (1.4) as being a discrete analoge of
the differential systems
x0 (t) = a(t)x + b(t)y
(1.5)
y 0 (t) = c(t)x + d(t)y
and
x0 (t) = a(t)x + b(t)y + f1 (t)
(1.6)
y 0 (t) = c(t)x + d(t)y + f2 (t)
respectively, where a, b, c, d, f1 , f2 are in C(R, R). If a(n) ≡ a, b(n) ≡ b, c(n) ≡ c
and d(n) ≡ d, then the characteristic equation of (1.3) is
λ2 − (a + c)λ + (ad − bc) = 0.
(1.7)
We note that this equation is the same for both the systems (1.3) and (1.5). Hence
the oscillatory behaviour of solutions of these systems are comparable. Clearly, the
components of the vector solution of (1.5) are oscillatory only if (1.7) has complex
roots. Otherwise, it is nonoscillatory. On the other hand, the behaviour of the
components of the vector solution of (1.3) is given below.
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OSCILLATION CRITERIA
3
Proposition 1.5. Let λ1 and λ2 be the roots of (1.7). If any one of the following
three conditions
(1) (a − d)2 + 4bc < 0,
1
(2) (a − d)2 + 4bc > 0 but (a + d) ± [(a − d)2 + 4bc] 2 < 0,
(3) (a − d)2 + 4bc = 0 and (a + d) < 0
hold, then every component of the vector solution of (1.3) is oscillatory. Otherwise,
there exists a nonoscillatory solution to (1.3).
The proof is simple and hence it is omitted.
The object of this work is to establish the sufficient conditions for the oscillation
of all solutions of the systems (1.3) and (1.4). Proposition 1.4 which demonstrate the
difference in the behaviour of the solutions of the systems (1.3)-(1.4) and (1.5)-(1.6)
motivate us to study further for the oscillatory behaviour of solutions of (1.3)-(1.4).
Furthermore, an attempt is made here to apply some of the results of [6] for the
oscillatory behaviour of solutions of the systems (1.3) and (1.4).
A close observation reveals that, all most all works in difference equations /
system of equations are the discrete analogue of the differential equations / system
of equations see for e.g. [1, 3, 4] and the references cited therein. Agarwal and Grace
[1] have discussed the oscillatory behavour of solutions of the system of equations
of the form
N
X
m+1 m
(−1)
∆ yi (n) +
qij yj (n − τjj ) = 0, m ≥ 1, i = 1, 2, . . . , N
j=1
which is the discrete analogue of the functional differential equations
N
X
dm
yi (t) +
qij yj (t − τjj ) = 0,
dtm
j=1
m ≥ 1, i = 1, 2, . . . , N,
where qij and τjj are real numbers and τjj > 0. It seems that the results in [1] are
the discrete analog results of the continuous case. We note that, in this work an
investigation is made to study the system of equations (1.3)/(1.4) without following
any results of the continuous case.
By a solution of (1.3)/(1.4) we mean a real valued vector function X(n) for n =
0, 1, 2 . . . which satisfies (1.3)/(1.4). We say that the solution X(n) = [x(n), y(n)]T
oscillates componentwise or simply oscillates if each component oscillates. Otherwise, the solution X(n) is called non-oscillatory. Therefore a solution of (1.3)/(1.4)
is non-oscillatory if it has a component which is eventually positive or eventually
negative.
We need the following two results from [6] for our use in the sequel.
Theorem 1.6. If an > 0, bn > 0 and
bn+1
bn
an ≤
+
an+1
an−1
for large n, then yn+2 − an yn+1 + bn yn = 0 is oscillatory.
Theorem 1.7. Let 0 ≤ an ≤ 1 and cn ≥ 0. Let fn = gn+2 − gn+1 , where for each
n ≥ 1, there exists m > n such that gn gm < 0. If
∞
∞
X
X
+
−
[(1 − an )gn+1
+ Cn gn+ ] = ∞,
[(1 − an )gn+1
+ Cn gn− ] = ∞,
n=1
n=1
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A. K. TRIPATHY
EJDE-2009/29
then all solutions of
yn+2 − an yn+1 + cn yn = fn
oscillate, where
gn+
= max{gn , 0} and gn− = max{−gn , 0}.
2. Oscillation for System (1.3)
Consider the system of equations (1.3) as
X(n + 1) = A(n)X,
T
where X(n) = [x(n), y(n)] and
a(n) b(n)
A(n) =
.
c(n) d(n)
We assume that a(n), b(n), c(n), d(n) are real valued functions defined for n ≥ n0 >
0. Let b(n) 6= 0 for all n ≥ n0 . Then it follows from (1.3) that
y(n) =
x(n + 1) a(n)
−
x(n);
b(n)
b(n)
that is,
y(n + 1) =
x(n + 2) a(n + 1)
−
x(n + 1).
b(n + 1)
b(n + 1)
Hence
c(n)x(n) + d(n)y(n) =
x(n + 2) a(n + 1)
−
x(n + 1);
b(n + 1)
b(n + 1)
that is,
x(n + 2) − P1 (n)x(n + 1) + Q1 (n)x(n) = 0,
(2.1)
where we define
P1 (n) = a(n + 1) +
Q1 (n) =
d(n)b(n + 1)
,
b(n)
b(n + 1)
[a(n)d(n) − b(n)c(n)]
b(n)
for all n ≥ n0 . Similarly, if c(n) 6= 0 for all n ≥ n0 , then
y(n + 2) − P2 (n)y(n + 1) + Q2 (n)y(n) = 0,
(2.2)
where we define
P2 (n) = d(n + 1) +
Q2 (n) =
a(n)d(n)
,
c(n)
c(n + 1)
[a(n)d(n) − b(n)c(n)]
c(n)
Theorem 2.1. Let Pi (n) > 0, Qi (n) > 0, i = 1, 2 be such that
Pi (n) ≤
Qi (n + 1)
Qi (n)
+
Pi (n + 1)
Pi (n − 1)
for all large n, then every solution X(n) of (1.3) oscillates.
(2.3)
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OSCILLATION CRITERIA
5
Proof. Suppose, on the contrary, that X(n) is a nonoscillatory solution of (1.3).
Then there exists n0 > 0 such that at least one component of X(n) is nonoscillatory for n ≥ n0 . Let x(n) be the nonoscillatory component of X(n) such that
x(n) is eventually positive for n ≥ n0 . Then applying Theorem 1.5, we have a
contradiction to (2.3). Similarly, one can proceed for y(n), if we assume that y(n)
is a nonoscillatory component of X(n) for n ≥ n0 . Hence the proof is complete. Remark. If (2.3) holds for either i = 1 or i = 2, then there could be a possibility
for the existence of nonoscillatory solution. However, we are not sure about the
fact. We note that (2.1) and (2.2) are non self-adjoint difference equations. Hence
the above possibility may be true.
Remark. If Pi (n) = pi and Qi (n) = qi , i = 1, 2 then (2.3) becomes p2i ≤ 2qi ,
i = 1, 2. Hence the inequalities p21 ≤ 2q1 and p22 ≤ 2q2 reduce to (a+d)2 ≤ 2(ad−bc).
Thus we have the following corollary.
Corollary 2.2. If A(n) ≡ A and (tr A)2 ≤ 2 det A, then (1.3) is oscillatory.
Example. Consider
x(n + 1)
1 −1 x(n)
(2.4)
=
y(n + 1)
2 1
y(n)
√
√
Indeed, tr A = 2 and det A = 3. λ1 = 1+i 2 and λ2 = 1−i 2 are two characteristic
roots of the coefficient matrix A. Clearly,
1
x(n) = λn1 √
i 2
√ n 1
√
= (1 + i 2)
i 2
1
n/2
√
= 3 (cos nθ + i sin nθ)
i 2
n/2
3 (cos nθ + i sin nθ
=
−3n/2 (sin nθ − i cos nθ)
and
1√
−i 2
√
1√
= (1 − i 2)n
−i 2
y(n) = λn2
1√
= 3n/2 (cos nθ − i sin nθ)
−i 2
n/2
3 (cos nθ + i sin nθ
= n/2
,
3 (sin nθ − i cos nθ)
√
where θ = tan−1 ( 2). By Corollary 2.2, the system (2.4) is oscillatory.
r(n)
t(n)
r(n)
If we define a(n) = r(n+1)
and d(n) = t(n+1)
, then r(n+1) = a(n)
and t(n+1) =
t(n)
d(n)
and hence solving the two relations we get
r(0)
r(n) = Qn−1
,
i=0 a(i)
d(0)
t(n) = Qn−1
,
j=0 d(j)
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A. K. TRIPATHY
EJDE-2009/29
where r(0) and d(0) are non-zero constants if a(n) 6= 0 6= d(n) for n ≥ n0 > 0.
From (1.3) it follows that
r(n + 1)x(n + 1) − r(n)x(n) = b(n)r(n + 1)y(n);
that is,
∆(r(n)x(n)) = b(n)r(n + 1)y(n).
Consequently,
n−1
X
∆[r(s)x(s)] =
s=0
n−1
X
b(s)r(s + 1)y(s);
s=0
that is,
x(n) =
=
n−1
1 X
r(0)x(0)
b(s)r(s + 1)y(s)
+
r(n)
r(n) s=0
n−1
Y
n−1
h
X b(s)y(s) i
Qs
a(i) x(0) +
.
i=0 a(i)
s=0
i=0
Similarly,
y(n) =
n−1
Y
n−1
h
X c(s)x(s) i
Qs
.
d(j) y(0) +
j=0 d(j)
s=0
j=0
Hence or otherwise the following theorem holds
Theorem 2.3. Let A(n) be a real valued coefficient matrix such that a(n) 6= 0 6=
d(n) for n ≥ n0 > 0. Then (1.3) is either oscillatory or nonoscillatory.
Theorem 2.4. Suppose that a(n) = 0 = d(n) and c(n) 6= 0 6= b(n) for all n ≥ n0 >
0. If lim inf n→∞ b(n) = α 6= 0 and lim inf n→∞ c(n) = β 6= 0 such that αβ < 0, then
(1.3) is oscillatory.
Proof. Let X(n) be a nonoscillatory solution of (1.3) for n ≥ n0 . Let x(n) be a
component of X(n) such that x(n) is eventually positive for n ≥ n0 . Clearly, from
(1.3) we obtain that, x(n) is a solution of
z(n + 2) − b(n + 1)c(n)z(n) = 0.
(2.5)
Without any loss of generality, we may assume that z(n) > 0 for n ≥ n0 . Equation
(2.5) can be written as
z(n + 2) z(n + 1)
= b(n + 1)c(n)
z(n + 1) z(n)
for n ≥ n0 . If we denote u(n) = z(n+1)
z(n) > 0 for n ≥ n1 , then the above equation
yields
lim inf [u(n + 1)u(n)] = lim inf [b(n + 1)c(n)]
n→∞
n→∞
(2.6)
= [lim inf b(n + 1)][lim inf c(n)] = αβ.
n→∞
n→∞
Since αβ 6= 0, then lim inf n→∞ [u(n)u(n+1)] exists. Let λ = lim inf n→∞ u(n). From
(2.6), it follows that f (λ) = λ2 −αβ = 0. It is easy to see that f (λ) attains minimum
at λ = 0. Consequently, min f (λ) ≤ f (λ) implies that αβ ≥ 0, a contradiction.
Hence (2.5) is oscillatory. Similarly, we can show that y(n) is a solution of
w(n + 2) − b(n)c(n + 1)w(n) = 0,
and (2.7) is oscillatory. This completes the proof.
(2.7)
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OSCILLATION CRITERIA
Example Consider the system of equations
x(n + 1)
0
−2 + (−1)n x(n)
=
,
y(n + 1)
2 + (−1)n
0
y(n)
7
n ≥ 0.
(2.8)
Indeed,
y(n + 2) + (5 − 4(−1)n )y(n) = 0,
n≥0
(2.9)
and α = −3, β = 1, αβ = −3 < 0. From Theorem 2.4, it follows that (2.8) is
oscillatory. We note that
n/2
y(n) = y(0)(−1)
n−2
Y
[5 − 4(−1)i ]
i=0
is one of the solution of (2.9), where (n/2) is an odd positive integer.
We conclude this section with the following result.
Theorem 2.5. Let X(n0 ) ∈ R × R for n0 ∈ Z + . If det A(n) 6= 0, then (1.3) is
Qn−1
oscillatory if and only if every component of the matrix i=n0 A(i) is oscillatory,
where
(
n−1
Y
A(n − 1)A(n − 2) . . . A(n0 ) n > n0
A(i) =
I
n = n0 .
i=n
0
The proof of the above theorem follows from the proof of the [3, Theorem 3.3]
and hence it is omitted.
Qn−1
Remark, If (1.3) is an autonomous system, then i=n0 A(i) = An−n0 and Theorem 2.5 holds for An−n0 for all n > n0 .
3. Oscillation for System (1.4)
This section presents sufficient conditions for the oscillation of all solutions of
the system of equations (1.4). If we assume that b(n) 6= 0 for all n ≥ n0 , then
y(n) =
f1 (n)
x(n + 1) a(n)
−
x(n) −
;
b(n)
b(n)
b(n)
that is,
y(n + 1) =
x(n + 2) a(n + 1)
f1 (n + 1)
−
x(n + 1) −
.
b(n + 1)
b(n + 1)
b(n + 1)
Consequently,
c(n)x(n) + d(n)y(n) + f2 (n) = y(n + 1)
implies that
x(n + 2) − P1 (n)x(n + 1) + Q1 (n)x(n) = G1 (n),
(3.1)
1 (n+1)
where G1 (n) = f2 (n) + fb(n+1)
, for n ≥ n0 and P1 (n), Q1 (n) are same as in (2.1).
Similarly, if we assume that c(n) 6= 0 for all n ≥ n0 , then we find
y(n + 2) − P1 (n)y(n + 1) + Q2 (n)y(n) = G2 (n),
(3.2)
f2 (n+1)
c(n+1) .
where P2 (n) and Q2 (n) are same as in (2.2) and G2 (n) = f1 (n) +
that Gi (n) could be oscillatory or could be nonoscillatory for i = 1, 2.
We note
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A. K. TRIPATHY
EJDE-2009/29
Theorem 3.1. Let Pi (n) < 0, Qi (n) > 0 for n ≥ n0 and i = 1, 2. Assume that
Gi (n) changes sign. In addition, there exists gi (n) which changes sign such that
Gi (n) = gi (n + 2) − gi (n + 1), i = 1, 2. If
∞
X
[Qi (n)gi+ (n) − Pi (n)gi+ (n + 1)] = ∞,
n=0
∞
X
[Qi (n)gi− (n) − Pi (n)gi− (n + 1)] = ∞
(3.3)
(3.4)
n=0
hold, then (1.4) is oscillatory, where
gi+ (n) = max{gi (n), 0}andgi− (n) = max{0, −gi (n)}
Proof. Suppose on the contrary that X(n) = [x(n), y(n)]T is a nonoscillatory solution of (1.4). Then there exists n0 > 0 such that at least one component of X(n)
is nonoscillatory for n ≥ n0 . Let x(n) be the nonoscillatory component of X(n)
such that x(n) > 0 for n ≥ n0 . Consequently, x1 (n) and x2 (n) are two solutions
of (3.1). Applying Theorem 1.6, we obtain a contradiction to our hypothesis (3.3).
A contradiction can be obtained to (3.4) if we assume that x(n) < 0 eventually for
n ≥ n0 . Similar observations can be dealt with the solution y(n) if we assume that
y(n) is a nonoscillatory component of X(n) for n ≥ n0 . Hence or otherwise the
proof of the theorem is complete.
Theorem 3.2. Let 0 ≤ Pi (n) < 1, Qi (n) > 0 and Gi (n) changes sign for i = 1, 2.
Assume that there exists gi (n) which changes sign such that Gi (n) = gi (n + 2) −
gi (n + 1), i = 1, 2. If
∞
X
[Qi (n)gi+ (n) + (1 − Pi (n))gi+ (n + 1)] = ∞,
n=0
∞
X
[Qi (n)gi− (n) + (1 − Pi (n))gi− (n + 1)] = ∞
n=0
hold, then (1.4) is oscillatory, where gi+ (n) and gi− (n) are same as in Theorem 3.1.
The proof of the above theorem follows from the Theorem 3.1 and Theorem 1.6
and hence it is omitted.
Theorem 3.3. Let Pi (n) < 0 and Qi (n) > 0 for all n ≥ n0 and i = 1, 2. Assume
that Gi (n) is nonoscillatory for all large n. Furthermore, assume that there exists
gi (n) such that Gi (n) = gi (n + 2) − gi (n + 1) and 0 < limn→∞ |gi (n)| < ∞. If
∞
X
[Qi (n)gi (n) − Pi (n)gi (n + 1)] = +∞,
(3.5)
n=0
∞
X
[Qi (n) − Pi (n)] = +∞
(3.6)
n=0
hold, then (1.4) is oscillatory.
Proof. Suppose on the contrary that X(n) = [x(n), y(n)]T is a nonoscillatory solution of (1.4). Proceeding as in the proof of the Theorem 3.1, we may assume that
x(n) and y(n) are nonosillatory solutions of (3.1) and (3.2) respectively. Assume
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9
that there exists n0 > 0 such that x(n) > 0 for n ≥ n0 . Then from (3.1), it follows
that
∆[x(n + 1) − g1 (n + 1)] = [P1 (n) − 1]x(n + 1) − Q1 (n)x(n) ≤ 0
(3.7)
but not identically zero for n ≥ n0 . Ultimately, (x(n+1)−gi (n+1)) is nonincreasing
on [n0 , ∞). We consider two cases viz. g1 (n) > 0 and g1 (n) < 0 for n ≥ n0 .
Suppose the former holds. If (x(n + 1) − g1 (n + 1)) > 0 for n ≥ n1 > n0 , then
limn→∞ (x(n + 1) − g1 (n + 1)) exists and hence (3.7) becomes
∞
X
[Q1 (n)g1 (n) − (P1 (n))g1 (n + 1)] < ∞,
n=n1
a contradiction to (3.5). Thus (x(n + 1) − g1 (n + 1)) < 0 for n ≥ n1 . Consequently,
x(n) < 0 for large n, a contradiction. Let the latter hold. Ultimately, x(n + 1) −
g1 (n + 1) > 0 for n ≥ n1 . It is easy to verify that 0 < limn→∞ x(n + 1) < ∞.
Let limn→∞ x(n) = `, ` ∈ (0, ∞). For every > 0, there exists n∗ > 0 such that
x(n + 1) > ` − > 0 for n ≥ n∗ .
Hence summing (3.7) from n2 to ∞, we get
∞
X
[Q1 (n)P1 (n)] < ∞, n2 > max{n1 , n∗ },
n=n2
a contradiction to our assumption (3.6). Same type of reasoning can be made if
we assume x(n) < 0 for n ≥ n0 . A similar type of observation can be formulated
when y(n) is a non-oscillatory component of (1.4) for n ≥ n0 . This completes the
proof.
Remark. Without any loss of generality, we may assume that gi (n) > 0 for i = 1, 2.
Theorem 3.4. Let 0 ≤ Pi (n) < 1 and Qi (n) > 0 for large n. Assume that all the
conditions of Theorem 3.3 hold except (3.5) and (3.6). If
∞
X
[Qi (n)gi (n) + (1 − Pi (n))gi (n + 1)] = ∞,
n=0
∞
X
[Qi (n) + (1 − Pi (n))] = ∞,
n=0
hold, then (1.4) is oscillatory.
The proof of the above theorem follows from the proof of the Theorem 3.3.
Example Consider
x(n + 1)
0
−1 x(n)
(−1)n
, n ≥ 0.
=
+
(−1)n
y(n + 1)
1/2 0
y(n)
Clearly, P1 (n) = 0 = P2 (n), Q1 (n) = 21 = Q2 (n), G1 (n) = 2(−1)n and G2 (n) =
(−1)n+1 . Indeed, x(n) and y(n) are two solutions of
1
z(n + 2) + z(n) = 2(−1)n ,
(3.8)
2
1
w(n + 2) + w(n) = (−1)n+1
(3.9)
2
respectively. If we choose g1 (n) = (−1)n and g2 (n) = 21 (−1)n+1 , then G1 (n) =
2(−1)n and G2 (n) = (−1)n+1 for all n ≥ 0.
10
A. K. TRIPATHY
EJDE-2009/29
It follows that all the conditions of Theorem 3.2 are satisfied and hence the given
system of equations is oscillatory. In particular, x(n) = 34 (−1)n is a solution of (3.8)
and y(n) = 32 (−1)n+1 is a solution of (3.9).
Example. Consider
x(n + 1)
−1 1
x(n)
1 − 2(−1)n
=
+
, n ≥ 0.
y(n + 1)
1 −2 y(n)
1 − 2(−1)n
where P1 (n) = −3 = P2 (n), Q1 (n) = 1 = Q2 (n), G1 (n) = 2 = G2 (n). Clearly, x(n)
and y(n) are solutions of
z(n + 2) + 3z(n + 1) + z(n) = 2,
(3.10)
w(n + 2) + 3w(n + 1) + w(n) = 2
(3.11)
respectively. If we choose g1 (n) = 2(n − 1) = g2 (n), then G1 (n) = 2 = G2 (n) and
hence (3.5) and (3.6) hold good. But we can not apply the Theorem 3.3 due to the
fact that
lim inf g1 (n) = lim sup g1 (n) = ∞.
n→∞
n→∞
√ √ n
3+ 5 n
2
n
(−1) is a solution of (3.10) and y(n) = 52 + 3+2 5 (−1)n
Then x(n) = 5 + 2
is a solution of (3.11). We note that the given system of equations is oscillatory.
Remark. In view of the above example, it seems that some additional condition
is necessary to prove the Theorem 3.3 when limn→∞ |gi (n)| = ∞.
Let a(n) = 0 = d(n) for all n ≥ n0 ≥ 0. Then the system of equations (1.4)
becomes
x(n + 1) = b(n)y(n) + f1 (n),
y(n + 1) = c(n)x(n) + f2 (n)
Solving the above two equations, it follows that x(n) and y(n) are solutions of
z(n + 2) − c(n)b(n + 1)z(n) = E1 (n),
(3.12)
w(n + 2) − c(n + 1)b(n)w(n) = E2 (n)
(3.13)
respectively, where E1 (n) = f1 (n+1)+f2 (n)b(n+1), E2 (n) = f2 (n+1)+f1 (n)c(n+
1) and we assume that det A(n) 6= 0 for all n ≥ n0 ≥ 0.
Theorem 3.5. Assume that c(n)b(n + 1) < 0 for all large n. If there exists ei (n),
i = 1, 2 which changes sign such that Ei (n) = ∆ei (n + 1) and
∞
X
+
[c(n)b(n + 1)e+
1 (n) − e1 (n + 1)] = −∞,
n=0
∞
X
+
[b(n)c(n + 1)e+
2 (n) − e2 (n + 1)] = −∞
n=0
−
where e+
i (n) = max{ei (n), 0}, ei (n) = max{−ei (n), 0}, then (1.4) is oscillatory.
It is easy to verify that, (3.12) and (3.13) can be written as
∆[z(n + 1) − e1 (n + 1)] = c(n)b(n + 1)z(n) − z(n + 1),
(3.14)
∆[w(n + 1) − e2 (n + 1)] = c(n + 1)b(n)w(n) − w(n + 1)
(3.15)
respectively. To prove this theorem it is sufficient to prove that (3.14) and (3.15)
are oscillatory. Moreover, the proof of the theorem can be done as in Theorems 3.1
and 1.6.
EJDE-2009/29
OSCILLATION CRITERIA
11
Remark. Ei (n) could be nonoscillatory also. If ei (n) is nonoscillatory such that
Ei (n) = ∆ei (n + 1), then a result corresponding to the Theorem 3.3 can be formulated under the conditions 0 < limn→∞ |ei (n)| < ∞ and c(n)b(n + 1) < 0 for all
large n.
Concluding Remarks. In this work, specific results regarding the oscillatory behaviour of vector solutions of the systems (1.3) and (1.4) have been established under the criteria det A(n) 6= 0 subject to the fundamental matrix Φ(n)(det Φ(n) 6= 0).
Indeed, the discrete analog of a second order differential equation is not necessarily
a self adjoint difference equation. Since the work in [6] based on the oscillatory
behaviour of solutions of a non-self adjoint difference equation and the author has
followed the work of [6], then it follows that the present work is not the analog work
of continuous case. Hence the results developed here may initiate further study for
the system of equations (1.3)/(1.4).
Existence of nonoscillatory vector solution of (1.3)/(1.4) is not discussed in this
work. However, the same can be followed from [3] and [4].
It is interesting to apply this work to study the system of equations
X(n + 1) = A(n)h(X(n))
and
X(n + 1) = A(n)h(X(n)) + F (n),
where h ∈ C(R, R).
Acknowledgements. The author is thankful to the anonymous referee for their
helpful suggestions and remarks.
References
[1] R. P. Agarwal, S. R. Grace; The oscillation of systems of difference equations, Appl. Math.
Lett. 13 (2000), 1-7.
[2] W. S. Burnside, A. W. Panton; The Theory of Equations, S. Chand and Company Ltd., New
Delhi, 1979.
[3] S. N. Elaydi; An Introduction to Difference Equations, Springer - Verlag, New York, 1996.
[4] I. Gyori, G. Ladas; Oscillation Theory of Delay Differential Equations with Applications,
Clarendon Press, Oxford, 1991.
[5] W. G. Kelley, A. C. Peterson; Difference Equations: An Introduction with Applications, Academic Press, INC, New York, 1991.
[6] N. Parhi, A. K. Tripathy; Oscillatory behaviour of second order difference equations, Commu.
Appl. Nonlin. Anal. 6(1999), 79 - 100.
Arun Kumar Tripathy
Department of Mathematics, Kakatiya Institute of Technology and Science, Warangal506015, India
E-mail address: arun tripathy70@rediffmail.com